Apply Green's Theorem to evaluate the integral.
-$$\oint _ { C } ( 8 y + x ) d x + ( y + 3 x ) d y$$
$C$ : The circle $( x - 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 9$
A) $45 \pi$
B) $- 270$
C) $- 30$
D) $- 45 \pi$

Question

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve $C$ to a double integral over the plane region $D$ bounded by $C$. It states that if $C$ is a positively oriented, piecewise smooth, simple closed curve in the plane, and $D$ is the region bounded by $C$, then$$\oint_{C} (L dx + M dy) = \iint_{D} \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA$$where $L(x, y) = 8y + x$ and $M(x, y) = y + 3x$.First, compute the partial derivatives:$$\frac{\partial M}{\partial x} = 3, \quad \frac{\partial L}{\partial y} = 8.$$Substitute these into the formula:$$\iint_{D} \left( 3 - 8 \right) dA = \iint_{D} (-5) dA.$$The area $A$ of the circle with radius $r = 3$ (since the equation is $(x - 6)^2 + (y - 3)^2 = 9$, and $9 = 3^2$) is $\pi r^2 = \pi (3)^2 = 9\pi$.Thus, the double integral becomes:$$-5 \times 9\pi = -45\pi.$$

Apply Green's Theorem to Evaluate the Integral ∮C(8y+x)dx+(y+3x)dy\oint _ { C } ( 8 y + x ) d x + ( y + 3 x ) d y∮C​(8y+x)dx+(y+3x)dy

Apply Green's Theorem to Evaluate the Integral $\oint _ { C } ( 8 y + x ) d x + ( y + 3 x ) d y$